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Describing the path of a heat seeking particle

  1. Jun 20, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the path of a heat-seeking particle placed at (4,3,10) with a temperature field

    [tex]T(x,y,z) = 400 - 2x^2 - y^2 -4z^2[/tex]


    2. Relevant equations

    Formulas for directional derivative and gradient.

    3. The attempt at a solution

    At any point in space (x,y,z), the particle must be moving in the direction which causes the greatest increase in T. This direction is given by the direction of

    [tex]\nabla T = < -4x, -2y, -8z > [/tex].

    I do not know how to continue from here.
     
  2. jcsd
  3. Jun 20, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    The velocity vector points in the direction of [itex]\nabla T[/itex] and so must be some multiple, say "k", of it.

    That means you must have dx/dt= -4kx, dy/dt= -2ky, and dz/dt= -8kz for some number k.
    You also are given that x(0)= 4(0), y(0)= 3, z(0)= 10. You can solve each of those for x, y, and z in terms of kt. That gives parametric equations for the path with parameter s= kt.
     
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